The Archimedean Solids & Their Dual Catalan Solids

The Archimedean solids and their duals the Catalan solids are less well known than the Platonic solids. Whereas the Platonic solids are composed of one shape, these forms that Archimedes wrote about are made of at least two different shapes, all forming identical vertices. They are 13 polyhedra of this type. And since each solid has a ‘dual’ there are also 13 Catalan solids. There are 26 in total. If our two-dimensional letters of the alpha-bet were three dimensional forms, these solids would be a great representation of our language.

The Archimedean Solids

Five Archimedean solids (on the bottom of illustration above) are derived from the Platonic solids by truncating or ‘cutting off’ the corners of each form.

The Platonic Solids

The vertices of the dual of a solid are the midpoints of the faces of the original solid. Among the Platonic solids, the octahedron and cube are duals of each other. The dodecahedron is the dual of the icosahedron. The last of these five solids is the tetrahedron which is a dual of itself when turned 180 degrees.

The Catalan Solids are the duals of the Archimedean solids. Named after belgian mathematician Eugène Catalan in 1865, these are made by placing a point in the middle of the faces of the Archimedean Solids and joining the points together with straight lines.

The Catalan Solids

The Archimedean and dual Catalan Solids

The number below each solid shows the sum of the angles on its surface. Since the cuboctahedron (in blue and purple on the left) is composed of 8 triangles and 6 squares, its surface contains a total of 3600°. Each triangle is made of 180° and each square 360°. (180° x 8) + (360° x 6) = 3600°

If found it interesting that 4 of the 13 Archimedean solids have surfaces that sum 7920°. 7920 is a number that seems to pop up frequently, not only as the mean diameter of earth in miles (credit: Not AScientist Around), but also the total number of protons & neutrons used by the 64 Codons that make up our DNA (credit: Anthony Morris). 7920 is also encoded into our units of measure, since 7920 inches = 1 furlong, and 7920 feet is exactly a mile and a half. 792 is also found in the bible as the diameter of New Jerusalem, being exactly 7,920,000 ft. (credit Randall Carlson).

The octahedron is to the Platonic solids; as the truncated cuboctahedron is to the Archimedean solids. The total angular measure of the Platonic solids equals 14,400° with the octahedron being exactly 1/10th of this number (1440°). The same is true for the truncated cuboctahedron (16560°), when compared to the total amount of degrees in the Archimedean solids (165600°).

Base-10 in the Platonic Solids

Base-10, and the recurrence of ‘ten’ are evident in the Platonic solids. There are exactly 5 x 10 sides among them, and 9 x 10 edges. If we contrive this base 10 symbolism a bit we can find 200, as the total number of sides, edges, points, and shapes used to make these 3-dimensional forms.

Ten is more than a number, it’s a concept for organizing reality. In other words, ten is not ‘ten’ because man decided that’s when we are going to stop using the single digit (9), or single glyph, or single shape, and add this concept of nothing (zero) behind it in order to create fractal scalability and retractability.

The Archimedean solids are further evidence of the way nature operates by using these canonical numerical archetypes to conceptually create what we know as reality, numerically, geometrically, musically, and cosmically.

“A man should hear a little music, read a little poetry, and see a fine picture every day of his life, in order that worldly cares may not obliterate the sense of the beautiful which God has implanted in the human soul.” ― Johann Wolfgang von Goethe

i love this page this is well a naturalist page it exacts my belief in a higher power- and intelligent design..i cut my apple first by slicing down to skin of the core then i use my index finger to form a equilateral triangle over the stem naval leveling off in cut direction i do the same only, i use for the three additional cuts is right triangle and what is left is a perfect wedge

Cosmic Creations Collection

“The object of life is not to be on the side of the majority, but to escape finding oneself in the ranks of the insane.” – Marcus Aurelius

“When truth is discovered by someone else, it loses something of its attractiveness.” -Aleksandr Solzhenitsyn

“All usurers are thieves and belong in the gallows.” -Martin Luther

“Those who love wisdom must investigate many things” – Heraclitus

“Don’t let schooling interfere with your education.” -Mark Twain

“Historians are dangerous people. They are capable of upsetting everything. They must be directed.” -Nikita Khrushchev

“Rise like lions after slumber, in unvanquishable number, shake your chains to earth like dew, which in sleep had fallen on you. Ye are many and they are few!” – Percy Shelly

“To learn who rules over you, simply find out who you are not allowed to criticize” -unknown

“If you want to be free, be moral” – St Augustine

“I have learned, that if one advances confidently in the direction of his dreams, and endeavors to live the life he has imagined, he will meet with a success unexpected in common hours.” ~ Henry David Thoreau

“Never refuse to see what you do not want to see, or what might go against your own cherished hypothesis or against the views of authorities. These are just the clues to follow up, as is also, and emphatically so, the thing you have never seen or heard before. The thing you cannot get a pigeon hole for is the finger point showing the way to discovery.” -Sir Patrick Manson (1844-1922)

“Out of the corruption of women proceeds the confusion of castes; out of the confusion of castes proceeds the loss of memory, out of the loss of memory proceeds the loss of understanding, and out of this all evil.”. -Bhagavad Gita,1:41

“Fever is Nature’s engine which she brings into the field to remove her enemy.” -Thomas Sydenham (1624-1689)

“Foolish the doctor who despises the knowledge acquired by the ancients.” ~Hippocrates

“Heretics are the only remedy against the entropy of human thought.” —Yevgeny Zamyatin

“You must unlearn what you have learned” -Yoda

Am I therefore become your enemy, because I tell you the truth? Galatians 4:16

“Nothing in this world is harder than speaking the truth, nothing easier than flattery.” ― Fyodor Dostoyevsky

Woe unto you, scribes and Pharisees, hypocrites! for ye are like unto whited sepulchres, which indeed appear beautiful outward, but are within full of dead men’s bones, and of all uncleanness. Even so ye also outwardly appear righteous unto men, but within ye are full of hypocrisy and iniquity. Matthew 23:27-28

“For nothing is secret, that shall not be made manifest; neither any thing hid, that shall not be known and come abroad.” Luke 8:17

“He who would enter into the realm of Wisdom must first divest himself of all intellectual pride. Prejudices, pre-conceived opinions and beliefs always stand in the way of true wisdom. Conceited opinions are always suicidal in their influences. They bar the door to the entrance of Truth.” -Ralph Waldo Trine

“Give instruction to a wise man, and he will be yet wiser:teach a just man, and he will increase in learning.” -Proverbs 9:9

“The Great Pyramid was a treasury of Divinely given wisdom embodying chronological, meteorological, astronomical, mathematical, historical and Biblical truth. ..This storehouse of wisdom remained sealed by Divine appointment, to be revealed to those now living; to whom these truths would bear witness, at a time when they would be most needed.” -E. Raymond Capt

“Our eyes are holden that we cannot see things that stare us in the face until the hour arrives when the mind is ripened. Then we behold them, and the time when we saw them not is like a dream.” ― Ralph Waldo Emerson

“The best way to make your dreams come true is to wake up.” ~ Paul Valery